3. DARK MATTER: BOUND TO GALAXIES OR SMOOTHLY DISTRIBUTED?

Given that the total masses of clusters like Coma greatly exceed their
luminous masses - although by amounts that are still fairly uncertain
- it is natural to wonder whether the d dark matter is bound to the
bright galaxies (in the
form of massive halos, say) or whether it is more smoothly
distributed. Here it is important to remember that there is currently no
strong evidence for massive halos around "hot" stellar systems, and thus
no reason to assume that the cluster dark matter is bound to galaxies,
most of which are elliptical. The fraction of cluster mass that is
associated with galaxies has important consequences for the rate of
galaxy-galaxy interactions and orbital decay (see below). One of the
first attempts to deal with this question was made by
Noonan (1970),
who noted that the tidal field produced by the potential well of a
cluster might be sufficiently strong to pull luminous matter
(along with, presumably, any dark-matter halos) from the brightest
galaxies. Noonan's hypothesis has been unaccountably neglected; most
subsequent work has focused instead on collisions as the physical
process that determines galaxy masses in clusters.

There are two major sources of uncertainty that make it difficult to
assess the importance of Noonan's tidal truncation hypothesis. 1. As
discussed above, neither the (macroscopic) distribution of the cluster
dark matter, nor the distribution of galaxy orbits, is well constrained,
even in the best-observed clusters like Coma. 2. Although simple tidal
theory makes a definite prediction about the limiting radius of a
galaxy, with specified mass, as it orbits about a potential
center, it says essentially nothing about the tidally-limited mass
itself. This is because the total mass depends strongly on the
distribution of matter near the galaxy's Lagrangian radius, and this
distribution is essentially unknown. A straightforward analysis gives a
feeling for the uncertainties involved. Consider a galaxy orbiting at a
fixed radius r0 from the center of a cluster. The
equations of motion, in the rotating frame, of a star that remains close
to the galaxy in the plane of its orbit are

(11a)

(11b)

Here =
(x, y) is the position of the star relative to the
galaxy center, with x parallel to the orbital radius vector
r0;
g() and
cl(r)
are the galaxy
and cluster potentials, respectively; and
is the angular
frequency of the galaxy orbit. Neglecting Coriolis terms, equation (11a)
predicts that a star would feel
a zero net force at the Lagrangian radius
L, where
L[2 -
(d2cl /
dr2)r0] =
(dg
/ d)L.
In order to derive a tidally-truncated mass from this equation, we need
to assume some relation between the (truncated) mass mg and the tidal
radius rt of the galaxy. Suppose that
Gmg =
2g2rt / 2, where
g is the galaxy
central velocity dispersion, and
is an unknown parameter
that specifies the shape of the halo density profile. If the dark matter
producing the tidal field is distributed roughly "isothermally" near the
cluster center, with central density
0,
then 2d2cl /
dr2G0, and

(12)

A more careful calculation would give a value (dependent on orbital
parameters) for the undetermined coefficient in equation
(12). Note, however, that the predicted mass depends sensitively
on the unknown structural parameter
. If the halo of the
tidally truncated galaxy has the same mass distribution as an
isotropic Michie-King model of high central concentration - a natural
assumption, given that these models look so much like globular clusters,
which are thought to be tidally truncated - then
1. But even along the
Michie-King sequence of models, a oscillates
between ~ 0.85 and ~ 1.3, giving an uncertainty of nearly a
factor of four to the predicted mass. Other reasonable families of
models (eg.
Wilson [1975]
spheres) have halos that look very different. Thus, simple tidal theory
can give only an order-ofmagnitude indication of the mass that could
remain bound to a galaxy in a cluster, even a cluster with a known
potential.

The only way to reduce this uncertainty is to simulate the tidal
truncation process via an N-body code.
Figure 4 shows the final masses
of model galaxies, containing 2000 particles initially, after orbiting
for a time ~ 40(G0)-1/2 about the center of a cluster
with mass density

(13)

(see Merritt and White
1987
for a description of the technique). For Zwicky's Coma model,
0 1 ×
10-2h2M
pc-3, making the simulated evolution time
~ 7 × 109h-1 years. According to
Figure 4,
galaxies on elongated, high-energy orbits are the most strongly
truncated, although the dependence of final mass on orbital parameters
is not great. Final masses of galaxies with pericenters near
rc
(the dark matter "core radius") are given approximately by

(14)

Detailed examination of the final states of these "truncated galaxies"
reveals that they contain about four times as much mass within their
tidal radii as do
= 1 Michie-King
models. The final tidal and half-mass radii are

(15)

Equation (14) predicts that a bright galaxy orbiting near the center of
the Coma cluster could indeed retain a fairly massive
halo, similar in
mass to the halos of bright spiral galaxies, as long as the central
density of dark matter in Coma is not much greater than in Zwicky's
model. Furthermore, equations (15) imply that little if any
luminous matter would be tidally removed, since characteristic
luminous radii of bright elliptical galaxies are
5h-1 kpc. Thus tidal
truncation is probably not a viable mechanism for producing the "diffuse
light" that is thought to be present in some clusters, including Coma
(e.g.
Thuan and Kormendy 1977).
Both of these conclusions would have to be modified if cluster central
densities could be shown to be much higher than in Zwicky's model. High
central densities are likely in clusters containing "cD" galaxies,
especially if the cD's are themselves surrounded by supermassive dark
halos. This point is discussed further in the next section.

Finally, we can estimate the total fraction of the dark matter in Coma
that could remain bound to galaxies in a model like Zwicky's. If we
assume a Schechter (1976)
galaxy luminosity function N(L)dL (L /
L*)-1 exp(- L /
L*) d (L /
L*), and furthermore that
the velocity dispersion of the halo material scales with the luminosity
of its parent galaxy as
g 225 km s-1
(L / L*)1/4, then equation
(14) predicts that a fraction

(16)

of the Coma cluster's mass should reside in galaxies. This
estimate is
an upper limit in the sense that it neglects the (probably greater, but
transient) tidal stresses during cluster formation, galaxy-galaxy
collisions, etc. It suggests that almost all of the dark matter in rich
clusters is smoothly distributed.